L(s) = 1 | + 2·3-s + 9-s + 11-s + 4·13-s − 6·19-s − 3·23-s − 4·27-s − 3·29-s + 2·33-s − 9·37-s + 8·39-s − 2·41-s − 9·43-s − 6·47-s − 6·53-s − 12·57-s − 8·59-s + 10·61-s + 67-s − 6·69-s − 7·71-s − 2·73-s − 9·79-s − 11·81-s + 12·83-s − 6·87-s + 4·89-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 1.37·19-s − 0.625·23-s − 0.769·27-s − 0.557·29-s + 0.348·33-s − 1.47·37-s + 1.28·39-s − 0.312·41-s − 1.37·43-s − 0.875·47-s − 0.824·53-s − 1.58·57-s − 1.04·59-s + 1.28·61-s + 0.122·67-s − 0.722·69-s − 0.830·71-s − 0.234·73-s − 1.01·79-s − 1.22·81-s + 1.31·83-s − 0.643·87-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.48135266333591152326887461839, −6.61347799900154443911151962880, −6.14986269829406337115742947794, −5.25018894835534068582117653206, −4.32564801139277232238843482887, −3.62932455379320392957589890551, −3.17996188305590154265545405661, −2.09039819645876920319023214642, −1.58540952942948094330610184498, 0,
1.58540952942948094330610184498, 2.09039819645876920319023214642, 3.17996188305590154265545405661, 3.62932455379320392957589890551, 4.32564801139277232238843482887, 5.25018894835534068582117653206, 6.14986269829406337115742947794, 6.61347799900154443911151962880, 7.48135266333591152326887461839